Download An Introductory Tutorial: Scott C Merrill October 10

An Introductory Tutorial:
Learning R for Quantitative Thinking in the Life Sciences
Scott C Merrill
October 10rd, 2012
Chapter 7 Distributions!
R has a function that allows one to generate samples based on distributions such as the normal
distribution. Compile the following to see some of the available distributions:
require(animation) # you will probably have to install this package
# sampling from a poisson distribution with different lambdas
# lambda = mean = standard deviation in a poisson distribution
# looping through lambdas from 1 to 20
#
with a 2 second pause between histograms
for (lam in 1:20) {
true.pop = rpois(n=10000,lambda = lam)
hist(true.pop)
ani.pause(2) # creates a pause between histogram displays (animation package)
}
# Generating samples from a Beta distribution
# looping through a beta distribution with different shape parameters
#
with shape parameters 1 & 2 ranging from 1:10
for (shp1 in 1:50) {
for (shp2 in 1:50) {
true.pop = rbeta(n=10000,shape1 = shp1/5, shape2 = shp2/5)
hist(true.pop, main = c("shape1 = ",shp1/5," shape2 =",shp2/5))
ani.pause(.1)
}
}
# Generating samples from a Weibull distribution
# looping through a weibull distribution with different shape parameters
#
with shape and scale parameters ranging from 1:10
for (shp1 in 1:50) {
for (scale1 in 1:50) {
Weibull = rweibull(n=10000,shape = shp1/5, scale = scale1/5)
hist(Weibull, main = c("shape = ",shp1/5," scale =",scale1/5))
ani.pause(.1)
}
}
# Now try a gamma distribution
for (shp1 in 1:50) {
for (scale1 in 1:50) {
Gamma.Distribution = rgamma(n=10000,shape = shp1/5, scale = scale1/5)
hist(Gamma.Distribution, main = c("shape = ",shp1/5,"
scale =",scale1/5))
ani.pause(.05)
}
}
Simulation to obtaining probability density and quantile functions
We have been using the r –prefix which samples from the distribution that you have described
following the r. For example rnorm generates samples from the normal distribution with a mean
and standard deviation defined by the user.
The “d” prefix:
Alternatively, one could find the probability density function. For example dnorm, where instead
of inputting the number of samples to obtain, one would enter values, with the probability
density value returned. Try compiling:
dnorm(x = 3,mean = 5, sd = 1)
Hint: the x value is 2 standard deviations away from the mean. What would you expect the
probability density function to equal?
The “p” prefix:
One could find the cumulative density function using the “p” prefix. For example pnorm, where
instead of inputting the number of samples to obtain, one would enter values of interest, with the
cumulative density value returned. Try compiling:
pnorm(q = 7, mean = 5, sd = 1)
Hint: the q value is 2 standard deviations above the mean. What would you expect the
cumulative density function to equal?
The “q” prefix:
One could find the value of the quantile using the “p” prefix. For example qnorm, where instead
of inputting the number of samples to obtain, one would enter cumulative probability of interest
and have the corresponding x value returned. Try compiling:
qnorm(p= .1, mean = 5, sd = 1) # returns value of x with 10% of probabilities at or below
or
qnorm(p = .9, mean = 5, sd = 1) # returns value of x with 90% of probabilities at or below
Do you understand what is happening here?
For those of you who are feeling ambitious, you can glance through Vito Ricci’s distribution
manuscript. It is fairly dense and has a lot of equations/jargon but has some really good
information on there:
http://cran.r-project.org/doc/contrib/Ricci-distributions-en.pdf
Exercise:
Simulate 10,50,200,1000,100000 data points in a sequence for a normal distribution with means
looping from -10 to 10, and standard deviation looping from 1 to 5 using the distribution code:
# code for generating 500 samples from a normal distribution with a mean of 100
#
and a standard deviation of 10 looks like
#
rnorm(n = 500,mean=100,sd=10)
#Create a histogram for each distribution (you do not have to output each of these plots, just
#
send me the code and the last plot (i.e., n=100000, mean = 10, sd = 5)
# put these code sections into a three piece nested loop with loops for sample number, mean
and standard deviation.
sample.number = c(10,50,200,1000,100000)
simulation.data = rnorm(n = x[loop.variable1], mean = [loop.variable2], sd = [loop.variable3])
hist(simulation.data)
Hint: you can create a for loop that runs from -10 to 10 using the same code structure that we
have been using:
for (x in -10:10) {
do stuff in here
}
Exercise:
Simulate data and histograms using the following distributions:
# log normal distribution e.g.,
rlnorm(meanlog=.8,sdlog = .5, n = 1000)
# exponential
rexp(n = 10000, rate = 2)
# What is happening here?
# hint:
f(x) = λ*exp^(- λ x)
Also, (from Ally’s cheat sheet) you could check out these distributions:
rt(n, df) # ‘Student’ (t)
rf(n, df1, df2) # Fisher–Snedecor (F) (c2)
rchisq(n, df) # Pearson
rbinom(n, size, prob) # binomial
rgeom(n, prob) # geometric
rhyper(nn, m, n, k) # hypergeometric
rlogis(n, location=0, scale=1) # logistic
rlnorm(n, meanlog=0, sdlog=1) # lognormal
rnbinom(n, size, prob) # negative binomial
runif(n, min=0, max=1) # uniform
rwilcox(nn, m, n)